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Towards a probability-free theory of continuous martingales

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  • Vladimir Vovk
  • Glenn Shafer

Abstract

Without probability theory, we define classes of supermartingales, martingales, and semimartingales in idealized financial markets with continuous price paths. This allows us to establish probability-free versions of a number of standard results in martingale theory, including the Dubins-Schwarz theorem, the Girsanov theorem, and results concerning the It\^o integral. We also establish the existence of an equity premium and a CAPM relationship in this probability-free setting.

Suggested Citation

  • Vladimir Vovk & Glenn Shafer, 2017. "Towards a probability-free theory of continuous martingales," Papers 1703.08715, arXiv.org.
    • Handle: RePEc:arx:papers:1703.08715
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    References listed on IDEAS

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    1. Vladimir Vovk, 2009. "Continuous-time trading and the emergence of probability," Papers 0904.4364, arXiv.org, revised May 2015.
    2. Yuri Kabanov & Robert Liptser, 2006. "From Stochastic Calculus to Mathematical Finance. The Shiryaev Festschrift," Post-Print hal-00488295, HAL.
    3. Vladimir Vovk, 2012. "Continuous-time trading and the emergence of probability," Finance and Stochastics, Springer, vol. 16(4), pages 561-609, October.
    4. Vladimir Vovk & Glenn Shafer, 2016. "A probability-free and continuous-time explanation of the equity premium and CAPM," Papers 1607.00830, arXiv.org.
    5. Vladimir Vovk, 2016. "Getting rich quick with the Axiom of Choice," Papers 1604.00596, arXiv.org, revised Mar 2017.
    6. Nicolas Perkowski & David J. Promel, 2013. "Pathwise stochastic integrals for model free finance," Papers 1311.6187, arXiv.org, revised Jun 2016.
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